# Prime Factors

In this section we learn about prime factors and how to find a whole number's prime factors.

A prime factor of a number is a prime number, which is also a factor of that number.

Remember that a prime number $$p$$ is a whole number that has exactly two factors (a.k.a divisors):

• itself: $$p$$, and
• 1
For example, $$7$$ is a prime number as it only has two factors: $$1$$ and $$7$$.

The first few prime numbers are: $2,3,5,7,11,13,17,\dots$ Every whole number greater than, or equal to, $$2$$ has at least one prime factor.

### Definition - Prime Factor

Given a whole number $$m$$, a prime factor $$p$$ of $$m$$ is a factor of $$m$$ which is also a prime number.
For instance one of the prime factors of $$12$$ is $$3$$, as $$3$$ is a factor of $$12$$ and $$3$$ is a prime number; indeed we can write: $12 = 3\times 4$ so $$3$$ is a factor of $$12$$ and since $$3$$ is a prime number it is a prime factor of $$12$$.

### Method - Listing a Number's Prime Factors

Given a whole number $$m$$, we can list all of its prime factors, $$p_1,p_2,p_3,\dots$$, using the following two steps:

• Step 1: Write a list of the first few prime numbers: $$2, 3, 5, 7, 11, 13, 17, 19, \dots$$ .

• Step 2: make the list of prime factors using all the factors of $$m$$ you see in the list of prime numbers, written in step 1.
This two-step method is further explained in the following tutorial:

## Tutorial: Listing a Number's Prime Factors

In the following tutorial we learn how to find the prime factors of a whole number.

## Exercise 1

List all of the prime factors of each of the following whole numbers

1. $$6$$

2. $$30$$

3. $$12$$

4. $$42$$

5. $$98$$

1. The prime factors of $$6$$ are $$2$$ and $$3$$.

2. The prime factors of $$30$$ are $$2$$, $$3$$, and $$5$$.

3. The prime factors of $$12$$ are $$2$$ and $$3$$.

4. The prime factors of $$42$$ are $$2$$, $$3$$ and $$7$$.

5. The prime factors of $$98$$ are $$2$$ and $$7$$.

## Solution With Working

1. To find the prime factors of $$6$$, we follow our two-step method:

• Step 1: list the first few prime numbers: $2,3,5,7,11,13, \dots$

• Step 2: find all of the factors of $$6$$ inside the list, made in step 1:

Looking at the list, the only factors of $$6$$ are: $$2$$ and $$3$$.
• The prime factors of $$6$$ are therefore $$2$$ and $$3$$.

2. To find the prime factors of $$30$$, we follow our two-step method:

• Step 1: list the first few prime numbers: $2,3,5,7,11,13, \dots$

• Step 2: find all of the factors of $$30$$ that are in the list, written in step 1:

Looking at the list, the only factors of $$30$$ are: $$2$$, $$3$$, and $$5$$.
The prime factors of $$30$$ are therefore $$2$$, $$3$$, and $$5$$.

3. To find the prime factors of $$12$$, we follow our two-step method:

• Step 1: list the first few prime numbers: $2,3,5,7,11,13, \dots$

• Step 2: find all of the factors of $$12$$ contained in the list, written in step 1:

Looking at the list, the only factors of $$12$$ are: $$2$$ and $$3$$.
The prime factors of $$12$$ are therefore $$2$$ and $$3$$.

4. To find the prime factors of $$42$$, we follow our two-step method!

• Step 1: list the first few prime numbers: $2,3,5,7,11,13, \dots$

• Step 2: find all of the factors of $$42$$ contained in the list, written in step 1:

Looking at the list, the only factors of $$42$$ are: $$2$$, $$3$$ and $$7$$.
• The prime factors of $$42$$ are $$2$$, $$3$$ and $$7$$.

5. To find the prime factors of $$98$$, we follow our two-step method:

• Step 1: list the first few prime numbers: $2,3,5,7,11,13, \dots$

• Step 2: find all of the factors of $$98$$ contained in the list, written in step 1:

Looking at the list, the only factors of $$98$$ are: $$2$$ and $$7$$.
• The prime factors of $$98$$ are $$2$$ and $$7$$.

## Exercise 2

State whether each of the statements that follow is true or false.

1. $$4$$ is a prime factor of $$12$$.

2. $$5$$ is a prime factor of $$20$$.

3. $$11$$ is a prime factor of $$21$$.

4. $$2$$ and $$7$$ are both prime factors of $$28$$.

5. $$9$$ is a prime factor of $$27$$.

1. false

2. true

3. false

4. true

5. false

## Solution with Working

1. false: $$4$$ is a factor of $$12$$ but it isn't a prime number so it isn't one of $$12$$'s prime factor's.

2. true: $$5$$ is a prime factor of $$20$$ since $$5$$ is a prime number and $$5$$ is a factor of $$20$$, indeed: $$20 = 5\times 4$$.

3. false: $$11$$ is a prime number but it isn't a factor of $$21$$ so it isn't one of its prime factors.

4. true: both $$2$$ and $$7$$ are prime numbers and they are both factors of $$28$$. Indeed:
• $$28 = 2\times 14$$, and

• $$28 = 7\times 4$$.

5. false: $$9$$ is a factor of $$27$$ but $$9$$ isn't a prime number so it isn't a prime factor.

## Prime Factorization

All whole numbers can be written as a product of its prime factors.

### Method

The method we learn here is based-on the fundamental theorem of arithmetic. In essence, it states:

All whole numbers greater than, or equal to $$2$$, can be written as a product of its prime factors.

## Tutorial: Writing a Number as the Product of its Prime Factors

In the following tutorial we learn how to write a number as a product of prime factors, watch it now:

## Exercise 3

Write each of the following whole numbers as its product of prime factors:

1. $$36$$
2. $$90$$
3. $$196$$
4. $$900$$
5. $$384$$
6. $$3600$$
7. $$210$$
8. $$540$$
9. $$144$$
10. $$180$$

## Solution Without Working

We find the following results:

1. $$36 = 2^2\times 3^2$$
2. $$90 = 2\times 3^2\times 5$$
3. $$196 = 2^2\times 7^2$$
4. $$900 = 2^2\times 3^2 \times 5^2$$
5. $$384 = 2^7\times 3$$
6. $$3600 = 2^4 \times 3^2 \times 5^2$$
7. $$210 = 2\times 3\times 5\times 7$$
8. $$540 = 2^2 \times 3^3 \times 5$$
9. $$144 = 2^4 \times 3^2$$
10. $$180 = 2^2 \times 3^2 \times 5$$